Automatic insertion of a turbulence model in the

finite element discretization of the Navier–Stokes equations

by Christine Bernardi1, Tomas Chacon Rebollo1,2, Frederic Hecht1 and Roger Lewandowski3

Abstract: We consider the finite element discretization of the Navier–Stokes equationslocally coupled with the equation for the turbulent kinetic energy through an eddy viscosity.We prove a posteriori error estimates which allow to automatically determine the zonewhere the turbulent kinetic energy must be inserted in the Navier–Stokes equations andalso to perform mesh adaptivity in order to optimize the discretization of these equations.Numerical results confirm the interest of such an approach.

Resume: Nous considerons une discretisation par elements finis des equations de Navier–Stokes couplees localement avec l’equation de l’energie cinetique turbulente par une vis-cosite turbulente. Nous prouvons des estimations d’erreur a posteriori qui permettentde determiner automatiquement la zone ou l’energie cinetique turbulente doit etre insereedans les equations de Navier–Stokes et simultanement d’adapter le maillage pour optimiserla discretisation de ces equations. Des resultats numeriques confirment l’interet d’une telleapproche.

1Laboratoire Jacques-Louis Lions, C.N.R.S. & Universite Pierre et Marie Curie - Paris 6,

boıte 187, 4 place Jussieu, 75252 Paris Cedex 05, France.

bernardi@ann.jussieu.fr, hecht@ann.jussieu.fr2

Departamento de Ecuaciones Diferenciales y Analisis Numerico, Universidad de Sevilla,

Tarfia s/n, 41012 Sevilla, Spain.

chacon@numer.us.es3

IRMAR (U.M.R. 6625), Universite de Rennes 1,

Campus de Beaulieu, 35042 Rennes Cedex 03, France.

roger.lewandowski@univ-rennes1.fr

1. Introduction.

Let Ω be a connected bounded open set in Rd, d = 2 or 3, with a Lipschitz–continuousboundary ∂Ω. The following system models the stationary flow of a viscous incompressibleturbulent fluid

−div(ν(k)∇u

)+ (u · ∇)u+ grad p = f in Ω,

divu = 0 in Ω,

−div(α(k)∇k

)= ν(k) |∇u|2 − k

√k

`in Ω,

u = g on ∂Ω,

k = 0 on ∂Ω.

(1.1)

The unknowns are the velocity u, the pressure p and the turbulent kinetic energy k, whilethe data are the distribution f and the function g. The functions ν and α are positive andthe parameter `, which represents the mixing length scale, is also positive. We refer to [35,Chap. 4] for the derivation of such a model. The main difficulty for its analysis is due to thefact that the right-hand member of the third equation is a priori not more than integrable.However the existence of a solution for similar problems when the function ν is boundedis proved in [19] and [29]. When the function ν is not bounded, the existence result is alsoestablished in [28, Chap. 5] for a scalar equation by a renormalization argument and in[27] for problem (1.1) by a regularity argument. It is also established in [3] and [4] in thecase of two coupled fluids. Relying on the approach of [3] and [27], it can be checked thatproblem (1.1) admits a solution when the function ν satisfies some additional assumptions.

In this work, both for mathematical convenience and in order not to handle all diffi-culties together, we consider the simplified model

−div(ν(k)∇u

)+ (u · ∇)u+ grad p = f in Ω,

divu = 0 in Ω,

−α∆k = ν(k) |∇u|2 in Ω,

u = 0 on ∂Ω,

k = 0 on ∂Ω,

(1.2)

where α is now a positive constant. The replacement of nonhomogeneous boundary condi-tions by homogeneous ones, i.e., taking g equal to 0, is only aimed to avoid the technicaldifficulties of the Hopf lemma, see [20, Chap. IV, Lemma 2.3]. Note also [4, §2] that thereplacement of div

(α(k)∇k

)by α∆k comes from Kirchoff’s change of unknown. More-

over, as suggested in [35, Chap. 4] and [10], we assume that the function ν admits thesimple expansion

∀ξ ∈ R+, ν(ξ) = ν0 + ν1

√ξ, (1.3)

1

for positive constants ν0 and ν1. Indeed, in practical situations, ν1 is a function whichdepends on the mixing length ` which appears in (1.1) and is approximatively of the sameorder; so, it is not restrictive to assume that it is constant in the case of problem (1.2).

Following the recent works [5] and [30], we are interested in the finite element dis-cretization of the still rather complex system (1.2). Our approach relies on the followingremark: Both experiments of measurements and numerical simulations indicate that, ina large part of the domain Ω, the turbulent energy k is negligible. So, our idea is toreplace ν(k) by ν0 in all subdomains where the quantity ν1

√k can be neglected without

increasing the global discretization error. Thus, we work with a finite element discretiza-tion of the reduced problem only. We refer to J. Hoffman and C. Johnson [24][25] for asimilar approach but in the completely different framework of coupling the so-called DirectNumerical Simulation and Large Eddy Simulation algorithms.

Of course, the discretization of the reduced problem must be coupled with mesh adap-tivity since small turbulence scales cannot be captured where the mesh is too coarse. Themathematical arguments for an optimal choice of the model coupled with mesh adaptiv-ity relies on the a posteriori analysis of the discrete problem, and the main ideas for theautomatic coupling of models are due to M. Braack and A. Ern [8]. Proving a posterioriestimates for nonlinear problems also relies on the approach of J. Pousin and J. Rappaz[38]. In addition, we need to handle some specific difficulties to build the error indicatorsrelated to the finite element discretization which are due to the nonlinear nature of theturbulent viscosity. We use arguments due to R. Verfurth [41, Chap. 3] for that. Notethat, in any case, the deep links between the model and the mesh make the numericalanalysis of the discrete problem more complex. However the numerical experiments thatwe present justify the interest of solving only a reduced problem.

An outline of the paper is as follows:• The main ideas and the corresponding algorithm for the automatic insertion of theturbulence model are described in Section 2.• In Section 3, we recall the arguments for proving the existence of a solution to problem(1.2).• Section 4 is devoted to the analysis of a fixed reduced problem. A first a posterioriestimate is also derived.• In Section 5, we describe the finite element discrete problem based on the variationalformulation of the previous reduced problem. We prove the existence of a solution and apriori error estimates.• In Section 6, we introduce two different families of error indicators, the first one beinglinked to the reduction of the problem and the second one to the finite element discretiza-tion. We prove a posteriori upper and lower bounds for the global error as a function ofthese indicators.• In Section 7, we present some numerical experiments that are in good coherence withthe analysis and justify the interest of automatic modeling.• Some conclusions are given in Section 8.

2

2. The ideas for automatic modeling.

The aim of automatic modeling in the present framework is to exhibit a partition ofthe domain Ω without overlap

Ω = Ωt ∪ Ω` and Ωt ∩ Ω` = ∅, (2.1)

(where the indices t and ` stand for turbulent and laminar, respectively), such that thefinite element discrete problem associated with system (1.2) is solved with the viscosityν(k) replaced by a constant ν0 on Ω`. This partition is considered as optimal when thedomain Ω` is chosen such that• the total computation time is highly diminished,• the error is not significantly increased,in comparison with solving the discrete problem associated with the full system (1.2).Indeed, the turbulence zone is limited to a part of the domain in a large number of flowssuch as wakes, jets, boundary layers and so on.

From now on, system (1.2) with the viscosity ν(k) replaced by a constant ν0 on Ω` iscalled “reduced problem”, and its finite element discretization is called “reduced discreteproblem”.

We propose an iterative algorithm for the choice of Ω` and Ωt and of a correspondingtriangulation, which is based on two families of error indicators (such indicators are definedin (6.46) and (6.4)− (6.5)): The first family deals with the modelization error and is madeof error indicators ηmK for all elements K of the triangulation which are contained in Ω` andthe second one is made of the standard residual error indicators ηK related to the finiteelement discretization for all elements K of the triangulation. Note that an analogousstrategy is used in [24] and [25] for the a posteriori error control of specific quantities.

Remark 2.1. In contrast with many other problems of automatic modeling, see [8], thechoice of Ωt and Ω` cannot be made independently of the mesh adaptation. Indeed usingthe turbulence model in subdomains where the mesh is too coarse is meaningless sincesuch a mesh cannot capture the small turbulence scales. So, in our case, the two types ofadaptivity must be performed simultaneously.

Initialization step: We first fix a triangulation T 0h of the domain Ω such that the

distance of the data f to an appropriate finite element space relying on this mesh issmaller than a given tolerance η∗ (the importance of such a choice is brought to light in[17]). We take Ω0

t = ∅ and Ω0` = Ω. So we first solve the discrete Navier–Stokes equations

(without eddy viscosity), next the discrete equation on k. Note indeed that the first andsecond line in (1.2) on one hand, and the third line on the other hand are uncoupled hereand moreover that the third line is a linear problem. Thus we are in a position to computethe error indicators ηmK and ηK .

Adaptation step: We assume that a partition of Ω into two subdomains Ωnt and Ωn`satisfying (2.1) is known, together with a triangulation T nh . We compute the solution ofthe associated reduced discrete problem, next the corresponding error indicators ηmK andηK , together with the mean values ηmh of the ηmK and the mean value ηh of the ηK . Next,we perform adaptivity in four substeps.

3

1. Adaptivity due to modeling error.All K in T nh such that ηmK is ≥ ηmh (we recall that such K are contained in Ωn` ) are insertedin a new domain Ωn+1

t . More precisely, this new domain Ωn+1t is the union of Ωnt and of

these new K.

2. Decomposition regularization.We perform the following regularization: Any element K which is not imbedded in Ωn+1

t

but is surrounded by elements which are imbedded in Ωn+1t , now belongs to the new domain

Ωn+1t . We skip the details for the construction of Ωn+1

t and choose Ωn+1` such that (2.1)

holds.

3. Mesh refinement due to the change of decomposition.The triangulation is automatically refined in the elements which belong to Ωn+1

t and not toΩnt , in order to capture the new scales of turbulence: Each of these elements is divided intosmaller subelements according to the ratio ηmh /η

mK . This gives rise to a new triangulation

T n+1h .

4. Adaptivity due to discretization error.When associating with each K in T n+1

h which do not belong to T nh the ηK′ , where K ′

belongs to T nh and contains K, we are in a position to perform a standard finite elementadaptivity strategy: The diameter of a new element contained in K or containing K isproportional to hK times the ratio ηh/ηK , where hK denotes the diameter of K. We referto [18] among others for more details on this procedure, specially in dimension d = 3. Thisgives rise to the triangulation T n+1

h .

The adaptation step is of course iterated either a finite number of times or until bothquantities

maxK∈T n

h,K⊂Ωn

`

ηmK and maxK∈T n

h

ηK ,

become smaller than the tolerance η∗.

Remark 2.2. Observe that Ωnt is contained in Ωn+1t for all n. A more complex strategy

must be used when working with the time-dependent system analogous to (1.2). Indeed,some triangles in Ωnt must go to Ωn+1

` in order to handle vortices quickly moving from onepart of the domain to another one. However we do not consider this situation here.

The numerical experiments which are presented in Section 7 justify the interest of thisalgorithm. Note also that the analysis which is performed in Sections 4 to 6 is perfectlyadapted to it. In particular, it leads to the construction of appropriate error indicators ηmKand ηK which make it very efficient.

4

3. About the continuous problem.

Throughout this section, we make the following non restrictive assumptions on thefunction ν:• It is a positive function on R satisfying

∀ξ ∈ R, ν(ξ) ≥ ν0, (3.1)

for a positive constant ν0.• It is bounded on R, i.e.

∀ξ ∈ R, ν(ξ) ≤ ν2, (3.2)

for a positive constant ν2.• It is continuous on R.Note that assumption (3.2) can be avoided: Recent results prove that most of the propertiesbelow still hold in the two-dimensional scalar case [12] or when the function ν is concave[27], but without convection term. However we prefer to work with (3.2), since this doesnot induce any restriction of the model (in practical situations, k is always bounded).

We use the standard notation for the Sobolev spaces Wm,p(Ω) and Wm,p0 (Ω) when

m is a positive integer and 1 ≤ p ≤ +∞. Moreover, with 1 < p < +∞ and 1p + 1

p′ = 1,W−m,p

′(Ω) is defined as the dual space of Wm,p

0 (Ω) (see [1, Thm 3.8] for its characteriza-tion) and the corresponding duality pairing is denoted by 〈·, ·〉. We also need the spacesHs(Ω) for positive real numbers s. As usual, L2

0(Ω) stands for the space of functions inL2(Ω) with a null integral on Ω.

From now on, we fix a real number r > d and take r′ such that

1r

+1r′

= 1.

Thus, we consider the variational problem,

Find (u, p, k) in H10 (Ω)d × L2

0(Ω)×W 1,r′

0 (Ω) such that

∀v ∈ H10 (Ω)d,

∫Ω

ν(k)∇u : ∇v dx+∫

Ω

(u · ∇)u · v(x) dx

−∫

Ω

(div v)(x)p(x) dx = 〈f ,v〉,

∀q ∈ L20(Ω), −

∫Ω

(divu)(x)q(x) dx = 0,

∀χ ∈W 1,r0 (Ω), α

∫Ω

grad k · gradχdx =∫

Ω

ν(k) |∇u|2 χ(x) dx.

(3.3)

Indeed standard arguments yield that this formulation is fully equivalent to problem (1.2).Moreover the product ν(k) |∇u|2 belongs to L1(Ω) and it follows from the Sobolev imbed-ding theorem that χ belongs to L∞(Ω), so that the right-hand member of the third equationis well defined.

5

We first state an a priori estimate on the solutions of problem (3.3). Note that theestimate concerning u is obviously derived by taking v equal to u in the first line of(3.3). The estimate concerning p involves the standard inf-sup condition on the bilinearform: (v, q) 7→ −

∫Ω

(div v)(x)q(x) dx, see [20, Chap. I, Cor. 2.1]. We refer to [7] for theargument leading to the estimate concerning k.

Proposition 3.1. For any data f in H−1(Ω)d, any solution (u, p, k) of problem (3.3)satisfies the following bound

ν0 ‖u‖H1(Ω)d ≤ c ‖f‖H−1(Ω)d ,

‖p‖L2(Ω) ≤ c((1 +

ν2

ν0) ‖f‖H−1(Ω)d +

1ν2

0

‖f‖2H−1(Ω)d

),

α ‖k‖W 1,r′ (Ω) ≤ cν2

ν20

‖f‖2H−1(Ω)d ,

(3.4)

for a constant c only depending on Ω and r.

Proving the existence of a solution for problem (3.3) relies on a fixed point theorem.We refer to [29] for the details (see also [3] and [4] for analogous results in a more complexsituation). As already hinted, the next theorem requires assumptions (3.1) and (3.2) (see[27] for other cases) and also the continuity of the function ν.

Theorem 3.2. For any data f in H−1(Ω)d, problem (3.3) admits a solution (u, p, k) inH1

0 (Ω)d × L20(Ω)×W 1,r′

0 (Ω). Moreover this solution satisfies (3.4).

Theorem 3.2 provides the existence result for problem (3.3) in the case we are inter-ested in, i.e. when the viscosity ν is given by

∀ξ ∈ R, ν(ξ) = minν0 + ν1

√ξ+, ν2

, (3.5)

where ξ+ stands for maxξ, 0. Indeed, applying the maximum principle yields that k isnonnegative on Ω (see [28, Chap. 4, §4.4.3]). Moreover, it appears later on that, at leastin dimension d = 2, it is bounded. So this choice does not seem restrictive.

From now on, we assume that Ω is a polygon or a polyhedron. We intend to investigatethe regularity properties of the solution (u, p, k) in this case. The idea of the next proofis due to N.G. Meyers [34]. It relies on the well-known property that the solution of theStokes problem with smooth enough data belongs to H

32 (Ω)d×H 1

2 (Ω), see [22, §7.3.3] forinstance.

Proposition 3.3. There exists a real number q0 > 2 depending on the geometry of Ω andon the ratio ν2/ν0 such that, for any q, 2 < q ≤ q0, and for any data f in W−1,q(Ω)d, anysolution (u, p, k) of problem (3.3) belongs to W 1,q(Ω)d × Lq(Ω) ×W 2, q2 (Ω) and satisfies,for a constant cq only depending on Ω, q, ν0 and ν2,

ν0 ‖u‖W 1,q(Ω)d + ‖p‖Lq(Ω) + α ‖k‖W 2, q2 (Ω)

≤ cq(‖f‖W−1,q(Ω)d + ‖f‖2H−1(Ω)d

). (3.6)

Proof: Let S] denote the Stokes operator, i.e., the operator which associates with anydata F in H−1(Ω)d, the part u of the solution (u, p) of the Stokes problem−∆u+ grad p = F in Ω,

divu = 0 in Ω,u = 0 on ∂Ω.

(3.7)

6

When taking ν02 equal to ν0+ν22 , we observe that the first two lines in problem (1.2) can

equivalently be written as

u+ S](div((1− ν(k)

ν02)∇u)

)= S]

(f − (u · ∇)uν02

). (3.8)

Furthermore, it follows from the Sobolev imbedding theorem that the term (u · ∇)ubelongs to Lr(Ω) for all r < 2 in dimension d = 2 and r ≤ 3

2 in dimension d = 3. Usingthe fact that Lr(Ω) is imbedded in W−1,r∗(Ω) if and only if the dual space W 1,r∗′

0 (Ω) ofW−1,r∗(Ω) (with 1

r∗ + 1r∗′ = 1) is imbedded in Lr

′(Ω) (with 1

r + 1r′ = 1) combined with

the Sobolev imbedding theorem, we obtain that (u · ∇)u belongs to W−1,r∗(Ω) for allr∗ < +∞ in dimension d = 2 and r∗ ≤ 3 in dimension d = 3. The same arguments,combined with (3.4), also yield that, for such an r∗,

∥∥∥f − (u · ∇)uν02

∥∥∥W−1,r∗ (Ω)d

≤ c(‖f‖W−1,r∗ (Ω)d + ‖f‖2H−1(Ω)d

). (3.9)

Next, we observe that:1) The Stokes operator S] is continuous from H−1(Ω)d into H1

0 (Ω)d with norm ≤ 1 andalso from W−1,q](Ω)d into W 1,q]

0 (Ω)d for q] = 6− d, see [21] or [32] and [33]. If χ denotesthe norm of S] in the space of linear applications from W−1,q](Ω)d into W

1,q]0 (Ω)d, an

interpolation argument (see [31, Chap. 1, Th. 5.1] or [1, Thms 7.17 & 7.20]) yields thatS] is continuous from W−1,q(Ω)d into W 1,q

0 (Ω)d for all q, 2 ≤ q ≤ q], with norm ≤ χθ(q),where θ is a continuous increasing function on [2, q]], equal to zero in 2 and to 1 in q] (wedo not make it precise for simplicity).2) For any q, 2 ≤ q ≤ ∞, the norm of the divergence operator from Lq(Ω)d×d intoW−1,q(Ω)d is ≤ 1, and the same property holds for the norm of the gradient operator fromW 1,q(Ω)d into Lq(Ω)d×d.3) For any q, 2 ≤ q ≤ ∞, and any function k, the norm of the multiplication by 1 − ν(k)

ν02

from Lq(Ω)d×d into itself is smaller than |1− ν2ν02| = ν2−ν0

ν2+ν0.

It follows from the previous arguments that the operator in the left-hand side of (3.8) isan automorphism of W 1,q(Ω)d for all q ≤ q] such that

χθ(q)ν2 − ν0

ν2 + ν0< 1. (3.10)

The existence of a q0 > 2 satisfying this condition comes from the fact that the limit ofχθ(q) when q tends to 2 is equal to 1 and that the quantity ν2−ν0

ν2+ν0is < 1.

Next, for a q ≤ q] satisfying (3.10) and since the right-hand side of (3.8) belongs toW 1,q(Ω)d, the velocity u belongs to W 1,q(Ω)d and, owing to (3.9), satisfies (3.6). Then,grad p obviously belongs to W−1,q(Ω), so that p belongs to Lq(Ω) and also satisfies (3.6)thanks to the estimate on u and (3.9). Finally, k is the solution of a Laplace equationwith right-hand side in L

q2 (Ω), hence belongs to W 2,q[(Ω) with q[ = min q2 ,

43 (see [22,

Thm 4.3.2.4], [15, Th. 2] or [16, Cor. 3.10]) and satisfies (3.6).

Note that the results of Proposition 3.3 can be slightly improved when the domain Ωis convex or has a smooth boundary.

7

Remark 3.4. Similar arguments yield that, if the function ν belongs to W 1,∞(R) and ifthe data f are smooth enough, the part u of the solution (u, p, k) of problem (3.3) belongsto Hs+1(Ω)d for a real number s > 0 also depending on the geometry of Ω and on thefunction ν.

We now state and prove a uniqueness result, only in dimension d = 2. Indeed, itsanalogue in dimension d = 3 would require an assumption on the regularity of the solutionwhich is not likely.

Proposition 3.5. Assume that the function ν is Lipschitz-continuous with Lipschitz con-stant ν∗0 . In dimension d = 2, for any real number q > 2 and any data f in W−1,q(Ω)2

such that, for appropriate constants c∗1 and c∗2,

c∗1ν2

0

(‖f‖W−1,q(Ω)2 +‖f‖2H−1(Ω)2

)≤ 1,

c∗2ν∗0

αν20

(‖f‖W−1,q(Ω)2 +‖f‖2H−1(Ω)2

)2 ≤ 1, (3.11)

problem (3.3) admits at most a solution (u, p, k) in H10 (Ω)2 × L2

0(Ω)×W 1,r′

0 (Ω).

Proof: Without restriction, we assume that q ≤ q0 for the q0 introduced in Proposition3.3, and we denote by cq(f) the quantity (which appears in estimate (3.6))

cq(f) = cq(‖f‖W−1,q(Ω)2 + ‖f‖2H−1(Ω)2

). (3.12)

Let (u1, p1, k1) and (u2, p2, k2) be two solutions of problem (3.3). We set: u = u1 − u2,p = p1 − p2, k = k1 − k2. Next, we proceed in three steps.1) The pair (u, p) satisfies

−div(ν(k1)∇u

)+ grad p

= div((ν(k1)− ν(k2))∇u2

)+ (u2 · ∇)u2 − (u1 · ∇)u1 in Ω,

divu = 0 in Ω,u = 0 on ∂Ω.

Writing these equations in the form (3.8), we derive thanks to the same arguments as forProposition 3.3

ν0 ‖u‖W 1,q(Ω)2 ≤ c(‖div

((ν(k1)− ν(k2)

)∇u2

)‖W−1,q(Ω)2

+ ‖(u2 · ∇)u2 − (u1 · ∇)u1‖W−1,q(Ω)2).

It follows from the Lipschitz property of ν that

‖div((ν(k1)− ν(k2)

)∇u2

)‖W−1,q(Ω)2 ≤ ν∗0 ‖k‖L∞(Ω)‖∇u2‖Lq(Ω)2×2 .

Thus, applying estimate (3.6) to u2 and using the Sobolev imbedding of W 2, q2 (Ω) intoL∞(Ω), we derive

‖div((ν(k1)− ν(k2)

)∇u2

)‖W−1,q(Ω)2 ≤ c

ν∗0ν0cq(f) ‖k‖

W 2, q2 (Ω).

8

Similarly, we have

‖(u2 · ∇)u2 − (u1 · ∇)u1‖W−1,q(Ω)2 ≤ c2ν0cq(f) ‖u‖W 1,q(Ω)2 .

Combining all this yields

ν0 ‖u‖W 1,q(Ω)2 ≤ cν∗0ν0cq(f) ‖k‖

W 2, q2 (Ω)+ c

2ν0cq(f) ‖u‖W 1,q(Ω)2 .

Thus, the following estimate follows from the first condition in (3.11) (with c∗1 = 4c cq)

ν0

2‖u‖W 1,q(Ω)2 ≤ c

ν∗0ν0cq(f) ‖k‖

W 2, q2 (Ω). (3.13)

2) The function k is a solution of the Laplace equation−α∆k = ν(k1) |∇u1|2 − ν(k2) |∇u2|2 in Ω,k = 0 on ∂Ω.

So, similar arguments lead to the estimates

α ‖k‖W 2, q2 (Ω)

≤ c′ν∗0ν2

0

cq(f)2 ‖k‖W 2, q2 (Ω)

+2c′ν2

ν0cq(f) ‖u‖W 1,q(Ω)2 .

So, the second condition in (3.11) (with c∗2 ≥ c2 = 2c′c2q) yields

α

2‖k‖

W 2, q2 (Ω)≤ 2c′ν2

ν0cq(f) ‖u‖W 1,q(Ω)2 . (3.14)

3) Combining (3.13) and (3.14) leads to

ν0

2‖u‖W 1,q(Ω)2 ≤ c

ν∗0ν0cq(f)

4c′ν2

αν0cq(f) ‖u‖W 1,q(Ω)2 .

When the second condition in (3.11) holds (with c∗2 = maxc2, 16cc′ c2qν2ν0), u is zero.

Thus, it follows from (3.14) that k is zero and from the equation

∀v ∈ H10 (Ω)2,

∫Ω

(div v)(x)p(x) dx = 0,

combined with the already quoted inf-sup condition on this form, that p is zero. Thisconcludes the proof.

The first condition in (3.11) with q = 2 is very similar to the condition which issufficient for the uniqueness of the solution of Navier-Stokes equations, see [20, Chap. IV,Thm 2.2]. Still more than for the Navier-Stokes equations, the conditions in (3.11) are toorestrictive and will not be assumed in what follows. Moreover it can be noted that thefunction ν defined in (3.5) is not Lipschitz–continuous. However and without restriction,we can replace this function by, for a fixed positive parameter ε,

∀ξ ∈ R, ν(ξ) = minν0 + ν1

√ξε+, ν2

, (3.15)

where ξε+ stands for maxξ, ε, in order to recover the Lipschitz property.

9

4. The reduced problem.

We now assume that the domain Ω admits a partition into Ωt and Ω` satisfying (2.1).We suppose that both Ωt and Ω` are polygons or polyhedra with Lipschitz–continuousboundaries. The strategy for an automatic determination of Ωt and Ω` is described inSection 2.

We also assume that the function ν is given by (3.15). We introduce a modifiedviscosity function ν∗ defined by

∀ξ ∈ R, ν∗(x, ξ) =ν(ξ) = minν0 + ν1

√ξε+, ν2 for a.e. x in Ωt,

ν0 for a.e. x in Ω`,(4.1)

i.e.,∀x ∈ Ω, ∀ξ ∈ R, ν∗(x, ξ) = χΩt(x) ν(ξ) + χΩ`(x) ν0, (4.2)

where χΩt and χΩ` stand for the characteristic functions of Ωt and Ω`, respectively. Next,we consider the reduced problem

−div(ν∗(·, k∗)∇u∗

)+ (u∗ · ∇)u∗ + grad p∗ = f in Ω,

divu∗ = 0 in Ω,

−α∆k∗ = ν∗(·, k∗) |∇u∗|2 in Ω,

u∗ = 0 on ∂Ω,

k∗ = 0 on ∂Ω.

(4.3)

The same arguments as in Section 3 imply that this problem admits the equivalentvariational formulation, for some real number r > d and with 1

r + 1r′ = 1,

Find (u∗, p∗, k∗) in H10 (Ω)d × L2

0(Ω)×W 1,r′

0 (Ω) such that

∀v ∈ H10 (Ω)d,

∫Ω

ν∗(x, k∗)∇u∗ : ∇v dx+∫

Ω

(u∗ · ∇)u∗ · v(x) dx

−∫

Ω

(div v)(x)p∗(x) dx = 〈f ,v〉,

∀q ∈ L20(Ω), −

∫Ω

(divu∗)(x)q(x) dx = 0,

∀χ ∈W 1,r0 (Ω), α

∫Ω

grad k∗ · gradχdx =∫

Ω

ν∗(x, k∗) |∇u∗|2 χ(x) dx.

(4.4)

The a priori estimates on the solution (u∗, p∗, k∗) are also easily derived from the fact thatthe function ν∗ is bounded from above and from below by the same constants as ν.

10

Proposition 4.1. For any data f in H−1(Ω)d, any solution (u∗, p∗, k∗) of problem (4.4)satisfies the following bound

ν0 ‖u∗‖H1(Ω)d ≤ c ‖f‖H−1(Ω)d ,

‖p∗‖L2(Ω) ≤ c((1 +

ν2

ν0) ‖f‖H−1(Ω)d +

1ν2

0

‖f‖2H−1(Ω)d

),

α ‖k∗‖W 1,r′ (Ω) ≤ cν2

ν20

‖f‖2H−1(Ω)d ,

(4.5)

for a constant c only depending on Ω and r.

The function ν∗ is no longer continuous on Ω, however it belongs to L∞(Ω× R) andeven to L∞(Ω; C 0(R)). So a direct extension of the arguments in [29] leads to the existenceresult.

Theorem 4.2. For any data f in H−1(Ω)d, problem (4.4) admits a solution (u∗, p∗, k∗)in H1

0 (Ω)d × L20(Ω)×W 1,r′

0 (Ω). Moreover this solution satisfies (4.5).

Note that the arguments used for the proof of Proposition 3.3 only requires the bound-edness of ν. So the results of this proposition still hold with ν replaced by ν∗.

Proposition 4.3. For the real number q0 > 2 introduced in Proposition 3.3, for any q,2 < q ≤ q0, and for any data f in W−1,q(Ω)d, any solution (u∗, p∗, k∗) of problem (4.4)belongs to W 1,q(Ω)d × Lq(Ω)×W 2, q2 (Ω) and satisfies (3.6).

We now intend to prove a first a posteriori estimate concerning the distance between(u, p, k) and (u∗, p∗, k∗). To do this, we write both problems (3.3) and (4.4) in a differentformulation, which is also useful for the numerical analysis of the discretization. Let Sdenote the Stokes operator, i.e., the operator which associates with any data F inH−1(Ω)d,the part u of the solution (u, p) of the Stokes-like problem (the viscosity ν0 correspondsto the case where Ωt is empty)−ν0 ∆u+ grad p = F in Ω,

divu = 0 in Ω,u = 0 on ∂Ω,

(4.6)

and similarly let L denote the inverse Laplace operator, i.e., the operator which associateswith any data G in W−1,r′(Ω), the solution k of the Laplace equation−α∆k = G in Ω,

k = 0 on ∂Ω.(4.7)

For a real number ρ which is made precise later on, we set:

X = W 1,ρ0 (Ω)d ×W 1,ρ

0 (Ω), (4.8)

and assume that f belongs to W−1,ρ(Ω)d. Thus it follows from Proposition 3.3 that, forappropriate values of ρ, problem (3.3) can equivalently be written as: Find U = (u, k)T inX such that

F(U) = U +(S 00 L

)G(U) = 0, (4.9)

11

where G is given by

G(U) =(

div((ν0 − ν(k))∇u

)+ (u · ∇)u− f

−ν(k) |∇u|2). (4.10)

In an analogous way, problem (4.4) can equivalently be written as: Find U∗ = (u∗, k∗)T

in X such that

F∗(U∗) = U∗ +(S 00 L

)G∗(U∗) = 0, (4.11)

where G∗ is given by

G∗(U) =(

div((ν0 − ν∗(·, k))∇u

)+ (u · ∇)u− f

−ν∗(·, k) |∇u|2). (4.12)

Let D denote the differentiation operator with respect to U . In what follows, we areled to assume that DF(U) is an isomorphism of X . So, we consider the following problem,for any data (Φ,Ψ) in H−1(Ω)d ×W−1,r′(Ω),

Find (w, π, κ) in H10 (Ω)d × L2

0(Ω)×W 1,r′

0 (Ω) such that

∀v ∈ H10 (Ω)d,

∫Ω

ν(k)∇w : ∇v dx+∫

Ω

ν′(k)κ∇u : ∇v dx

+∫

Ω

(u · ∇)w · v(x) dx+∫

Ω

(w · ∇)u · v(x) dx

−∫

Ω

(div v)(x)π(x) dx = 〈Φ,v〉,

∀q ∈ L20(Ω), −

∫Ω

(divw)(x)q(x) dx = 0,

∀χ ∈W 1,r0 (Ω), α

∫Ω

gradκ · gradχdx = 〈Ψ, χ〉

+ 2∫

Ω

ν(k)∇u · ∇w χ(x) dx+∫

Ω

ν′(k)κ |∇u|2 χ(x) dx.

(4.13)

This problem makes sense when the mappings

v 7→∫

Ω

ν′(k)κ∇u : ∇v dx and χ 7→∫

Ω

ν′(k)κ |∇u|2 χ(x) dx,

are linear continuous forms on H10 (Ω)d and W 1,r(Ω), respectively. Since W 1,r(Ω) with

r > d is imbedded in L∞(Ω) and ν′ is bounded, this holds when κ∇u belongs to L2(Ω)d×d.It can be checked that this property is satisfied for any (u, κ) in W 1,q(Ω)3×W 2, q2 (Ω) whenq > 2 in dimension d = 2 and q ≥ 12

5 in dimension d = 3.

As a consequence, when taking ρ > 2 (for instance) in dimension d = 2 and ρ ≥ 125 in

dimension d = 3, the assumption that DF(U) is an isomorphism of X can equivalently bewritten as follows: For any data (Φ,Ψ) in W−1,ρ(Ω)d ×W−1,ρ(Ω), problem (4.13) has aunique solution and this solution belongs to W 1,ρ(Ω)d×Lρ(Ω)×W 1,ρ(Ω). It can be noted

12

that this assumption only requires the local uniqueness of the solution (u, p, k) and it iswell-known to be less restrictive than the conditions for its global uniqueness as given inProposition 3.5 (see [20, Chap. IV, §3.2] for the analogue for the Navier–Stokes equations).

We also need that the mapping: V 7→ DF(V ) is Lipschitz–continuous at least in aneighbourhood of U . This requires for instance that the mapping:

k 7→∫

Ω

ν′(k)κ∇u : ∇v dx,

is Lipschitz–continuous or, equivalently that the product ν′′(k)kκ∇u : ∇v is integrablefor all u in W 1,q(Ω)d and v in H1(Ω)d, k and κ in W 1,q(Ω). This is true when ν′′ isbounded and the product kκ belongs to Lt(Ω) with 1

t + 1q = 1

2 . This property is alwaysvalid in dimension d = 2 since W 1,q(Ω) is imbedded in L∞(Ω) for all q > 2, but onlyholds when q ≥ 18

7 in dimension d = 3. For simplicity, we have decided to make the nexthypothesis.

Assumption 4.4. In the case of dimension d = 3, there exists a real number q0 ≥ 3 suchthat, for any q, 2 < q ≤ q0, and for any data f in W−1,q(Ω)d, any solution of problem(3.3) or (4.4) belongs to W 1,q(Ω)d × Lq(Ω)×W 1,q(Ω).

When looking at (3.10), we observe that this assumption is satisfied at least when theratio ν2/ν0 is small enough. So, from now on, we take

2 < ρ < q0 in dimension d = 2 and ρ = 3 in dimension d = 3. (4.14)

We are now in a position to prove the a posteriori estimate. This relies on a result due toJ. Pousin and J. Rappaz [38] (see also [41, §2.1] for another statement). We introduce thefunction µ defined by

∀ξ ∈ R, µ(ξ) = minν1

√ξε+, ν2 − ν0, (4.15)

and, in analogy with (4.2), the function µ∗ defined by

∀x ∈ Ω,∀ξ ∈ R, µ∗(x, ξ) = χΩ`(x)µ(ξ). (4.16)

Proposition 4.5. If the function ν belongs to W 2,∞(R) and Assumption 4.4 is satisfied,let (u, p, k) be a solution of problem (3.3) such that DF(U), with U = (u, k)T, is anisomorphism of X . There exists a positive number R only depending on this solution suchthat the following a posteriori error estimate holds

‖u− u∗‖W 1,ρ(Ω)d + ‖k − k∗‖W 1,ρ(Ω)

≤ c(‖div

(µ∗(·, k∗)∇u∗

)‖W−1,ρ(Ω)d + ‖µ∗(·, k∗) |∇u∗|2‖W−1,ρ(Ω)

).

(4.17)

for any solution (u∗, p∗, k∗) of problem (4.4) such that (u∗, k∗) belongs to the ball of Xwith centre (u, k) and radius R.

13

Proof: Since ν belongs to W 2,∞(R), it follows from Assumption 4.4 and the choice(4.14) of ρ that the mapping F is continuously differentiable on X and moreover thatthe mapping: V 7→ DF(V ) is Lipschitz–continuous with values in the space of linearendomorphisms of X . Let λ denote the corresponding Lipschitz constant. Applying [41,Prop. 2.1] yields that there exist positive constants R and c only depending on the normof DF(U)−1 and λ such that the following estimate holds for any solution U∗ of problem(4.11) satisfying ‖U − U∗‖X ≤ R,

‖U − U∗‖X ≤ c ‖F(U∗)‖X = c ‖F(U∗)−F∗(U∗)‖X .

To evaluate this last quantity, we observe that

F(U∗)−F∗(U∗) = −(S 00 L

)(div((ν(k∗)− ν∗(·, k∗))∇u∗

)(ν(k∗)− ν∗(·, k∗)) |∇u∗|2

).

Since S maps W−1,ρ(Ω)d into W 1,ρ(Ω)d (see [32] and [33] in dimension d = 3) and L mapsW−1,ρ(Ω) into W 1,ρ(Ω) (see [16, Thm 1.1]), we derive

‖U − U∗‖X

≤ c(‖div (ν(k∗)− ν∗(·, k∗))∇u∗‖W−1,ρ(Ω)d + ‖(ν(k∗)− ν∗(·, k∗)) |∇u∗|2‖W−1,ρ(Ω)

).

The function ν(k∗) − ν∗(·, k∗) is equal to minν1

√k∗ε+, ν2 − ν0 = µ(k∗) on its support

which is contained in Ω`. So this gives the desired result.

In the previous proposition, we have made the assumption that the solution (u∗, p∗, k∗)of problem (4.4) is not too far from (u, p, k). Since the reduced problem (4.4) obviouslycoincides with problem (3.3) when Ωt is equal to Ω, we now state a convergence resultwhich makes this property consistent. For simplicity, we only give an abridged proof ofthe next proposition.

Proposition 4.6. Let (Ωtn)n be an increasing sequence of open polygons or polyhedrawith Lipschitz–continuous boundaries such that ∪n∈N Ωtn is equal to Ω, and let Ω`n beequal to Ω \ Ωtn. Let also (un, pn, kn) be a solution of problem (4.4) with Ωt = Ωtnand Ω` = Ω`n. If the function ν belongs to W 1,∞(R), Assumption 4.4 holds and thedata f belong to W−1,ρ(Ω)d ∩ Hs−1(Ω)d for a real number s, 0 < s < 1, there exists asubsequence (un′ , pn′ , kn′)n′ which converges to a solution (u, p, k) of problem (3.3) weaklyin W 1,ρ(Ω)d × Lρ(Ω)×W 1,ρ(Ω).

Proof: It follows from Proposition 4.3 that the sequence (un, pn, kn)n satisfies

‖un‖W 1,ρ(Ω)d + ‖pn‖Lρ(Ω) + ‖kn‖W 1,ρ(Ω) ≤ cρ(‖f‖W−1,ρ(Ω)d + ‖f‖2H−1(Ω)d

),

and also (see Remark 3.4), for a real number s0 > 0 small enough,

‖un‖Hs0+1(Ω)d ≤ cρ ‖f‖Hs0−1(Ω)d .

So, there exists a subsequence (un′ , pn′ , kn′)n′ which converges to (u, p, k) weakly inW 1,ρ(Ω)d × Lρ(Ω) ×W 1,ρ(Ω). It follows from compactness arguments that there exists

14

another subsequence, still denoted by (un′ , pn′ , kn′)n′ for simplicity, such that (un′ , kn′)n′converges to (u, k) strongly in H1(Ω)d × Lq(Ω) for an appropriate real number q > 2.From this, we deduce that

∀v ∈ H10 (Ω)d, lim

n′→+∞

∫Ω

ν∗(x, kn′)∇un′ : ∇v dx =∫

Ω

ν(k)∇u : ∇v dx,

∀χ ∈W 1,r0 (Ω), lim

n′→+∞

∫Ω

ν∗(x, kn′) |∇un′ |2 χ(x) dx =∫

Ω

ν(k) |∇u|2 χ(x) dx.

So, when passing to the limit in the problem satisfied by (un, pn, kn), it can be checkedfrom the previous convergence properties that (u, p, k) is a solution of problem (3.3).

Note to conclude that the function ν defined in (3.15) does not belong to W 2,∞(R).More precisely its derivative is continuous everywhere but at the two points ξ = ε andξ = ξ0 = (ν2−ν0ν1

)2. Without restriction, we now work with a regularization of the functionν which coincides with ν except in the neighbourhoods ] ε2 ,

3ε2 [ and ]ξ0 − ε

2 , ξ0 + ε2 [ of these

two points, as illustrated in Figure 1. We do not make precise this regularization forsimplicity and still denote the regularized function by ν.

1

Figure 1. The function ν and its regularization

15

5. The reduced discrete problem.

In view of the discretization, we introduce a regular family (Th)h of triangulations ofΩ by closed triangles (d = 2) or tetrahedra (d = 3), in the usual sense that• for each h, Ω is the union of all elements of Th,• for each h, the intersection of two different elements of Th, if not empty, is a corner, awhole edge or a whole face of both elements,• the ratio of the diameter hK of an element K in Th to the diameter of its inscribed circleor sphere is bounded by a constant σ independent of K and h.As standard, h denotes the maximum of the diameters of the elements of Th. We makethe further and non restrictive assumption that each element K of Th is contained eitherin Ωt or in Ω`. From now on, c, c′, . . . stand for generic constants which may vary fromline to line but are always independent of h.

For each nonnegative integer m and any K in Th, let Pm(K) denote the space ofrestrictions to K of polynomials with d variables and total degree ≤ m. As standard forthe Stokes problem, we have decided to work with the Taylor–Hood finite elements, see[26]. Consequently, the discrete spaces of velocities and pressures are defined by

Xh =vh ∈ H1

0 (Ω)d; ∀K ∈ Th, vh|K ∈ P2(K)d,

Mh =qh ∈ H1(Ω) ∩ L2

0(Ω); ∀K ∈ Th, qh|K ∈ P1(K).

(5.1)

We also use piecewise quadratic functions for approximating the turbulent kinetic energyk in order to preserve the convergence order equal to 2 for the previous elements. So thediscrete space of energies is defined by

Yh =χh ∈W 1,r

0 (Ω); ∀K ∈ Th, χh|K ∈ P2(K). (5.2)

The discrete problem is then built from (4.4) by the Galerkin method. It reads

Find (uh, ph, kh) in Xh ×Mh × Yh such that

∀vh ∈ Xh,

∫Ω

ν∗(x, kh)∇uh : ∇vh dx+∫

Ω

(uh · ∇)uh · vh(x) dx

−∫

Ω

(div vh)(x)ph(x) dx = 〈f ,vh〉,

∀qh ∈Mh, −∫

Ω

(divuh)(x)qh(x) dx = 0,

∀χh ∈ Yh, α

∫Ω

grad kh · gradχh dx =∫

Ω

ν∗(x, kh) |∇uh|2 χh(x) dx.

(5.3)

The imbeddings of Xh in H10 (Ω)d, of Mh in L2

0(Ω) and of Yh both in W 1,r0 (Ω) and in

W 1,r′

0 (Ω) yield that the discretization is fully conforming.

Remark 5.1. In the implementation of this problem, ν∗(·, kh) is replaced by the functionνh such that, for each K in Th, νh|K belongs to P2(K) and which is equal to ν∗(a, kh(a)) at

16

each vertex and midpoint a of an edge of K. We do not take into account this modificationin the analysis for simplicity.

The numerical analysis of this problem relies on the Brezzi–Rappaz–Raviart theorem,see [9] or [20, Chap. IV, Thm 3.1]. We recall that problem (4.4) admits the formulation(4.11); however we are led to use a modified formulation. We now work with X replacedby the space Y defined by

Y = H10 (Ω)d ×

(H1

0 (Ω) ∩ L∞(Ω)), (5.4)

and equipped with the corresponding norm, which seems more natural in view of the finiteelement discretization. For any real number or real-valued measurable function ξ on Ω,we introduce the Stokes operator S(ξ) which associates with any data F in H−1(Ω)d, thepart u of the solution (u, p) of the generalized Stokes problem

−div(ν∗(·, ξ)∇u

)+ grad p = F in Ω,

divu = 0 in Ω,u = 0 on ∂Ω.

(5.5)

We also use the operator L defined in (4.7). Thus, problem (4.4) can equivalently bewritten as: Find U∗ = (u∗, k∗)T in Y such that

F(U∗) = U∗ +(S(k∗) 0

0 L

)G(U∗) = 0, (5.6)

where G is given by

G(U) =(

(u · ∇)u− f−ν∗(·, k) |∇u|2

). (5.7)

Similarly, let Sh(ξ) denote the discrete Stokes operator associated with problem (5.5),i.e., the operator which associates with any data F in H−1(Ω)d, the part uh of the solution(uh, ph) in Xh ×Mh of the Stokes problem

∀vh ∈ Xh,

∫Ω

ν∗(x, ξ)∇uh : ∇vh dx−∫

Ω

(div vh)(x)ph(x) dx = 〈F ,vh〉,

∀qh ∈Mh, −∫

Ω

(divuh)(x)qh(x) dx = 0.(5.8)

Let finally Lh denote the discrete inverse Laplace operator, i.e., the operator which asso-ciates with any data G in H−1(Ω), the solution kh in Yh of the Laplace equation

∀χh ∈ Yh, α

∫Ω

grad kh · gradχh dx = 〈G,χh〉. (5.9)

Indeed, an inf-sup condition on the form: (vh, qh) 7→ −∫

Ω(div vh)(x)qh(x) dx between

the spaces Xh and Mh is proven for instance in [20, Chap. II, Cor. 4.1], so that theseoperators are well-defined. Problem (5.3) now admits the equivalent formulation: FindUh = (uh, kh)T in Y such that

Fh(Uh) = Uh +(Sh(kh) 0

0 Lh

)G(Uh) = 0. (5.10)

17

To go further, we recall from a simple extension of [20, Chap. II, Thm 4.3] that theoperator Sh(ξ) satisfies the following properties: For any F in H−1(Ω)d,

‖Sh(ξ)F ‖H1(Ω)d ≤ c ‖F ‖H−1(Ω)d , (5.11)

and, if moreover F belongs to Hs−1(Ω)d and S(ξ)F to Hs+1(Ω)d for a real number s,0 ≤ s ≤ 2,

‖(S(ξ)− Sh(ξ)

)F ‖H1(Ω)d ≤ c hs

(‖S(ξ)F ‖Hs+1(Ω)d + ‖F ‖Hs−1(Ω)d

). (5.12)

Moreover, since the function ν∗(·, ξ) is bounded from above and from below, both constantsin estimates (5.11) and (5.12) are independent of ξ. The analogous properties concerningthe operator Lh are also standard [6, Chap. X, Th. 1.1 & 1.2]: For any data G in H−1(Ω),

‖LhG‖H1(Ω) ≤ c ‖G‖H−1(Ω), (5.13)

and, if moreover LG belongs to Hs+1(Ω) for a real number s, 0 ≤ s ≤ 2,

‖(L − Lh)G‖H1(Ω) ≤ c hs ‖LG‖Hs+1(Ω). (5.14)

We refer to [36, Cor. 2] for the less usual result: For any data G in H−1(Ω) such that LGbelongs to W s+1,∞(Ω) for s equal to 0, 1 or 2,

‖(L − Lh)G‖L∞(Ω) ≤ c hs+1 ‖LG‖W s+1,∞(Ω). (5.15)

Moreover, we are led to assume that, if LG belongs to L∞(Ω),

‖LhG‖L∞(Ω) ≤ c ‖LG‖L∞(Ω). (5.16)

This property is proved in [39, Thm 5.1] only when the family of triangulations (Th)h isuniformly regular. However it seems possible to extend this result to more general familiesof triangulations.

Let us introduce the matrix operators

M(ξ) =(S(ξ) 0

0 L

), Mh(ξ) =

(Sh(ξ) 0

0 Lh

). (5.17)

The following convergence result is easily derived from (5.11) to (5.16) by a density argu-ment: When property (5.16) holds, for any (F , G) in H−1(Ω)d × H−1(Ω) such that LGbelongs to C 0(Ω),

limh→0

‖(M(ξ)−Mh(ξ)

)(F , G)‖Y = 0.

Moreover, we recall from [14, Cor. 18.15] that, since Ω is a polygon or a polyhedron, theinverse Laplace operator maps Ht(Ω) into Ht+2(Ω) for some real number t > d

2 − 2. Sincethis Ht+2(Ω) is compactly imbedded in C 0(Ω), we deduce from the previous property that

limh→0

sup(F ,G)∈K

‖(M(ξ)−Mh(ξ)

)(F , G)‖Y = 0, (5.18)

18

for any bounded set K of the space

Y = Hτ (Ω)d ×Ht(Ω), −1 < τ < 0,d

2− 2 < t < 0.

Let also E(Y) denote the space of endomorphisms of Y. We are now in a position toprove the following lemmas.

Lemma 5.2. Let (u∗, p∗, k∗) be a solution of problem (4.4) such that U∗ = (u∗, k∗)T

belongs to Hs+1(Ω)d×W 1,r(Ω), s > d−2 and r > d, and that DF(U∗) is an isomorphismof Y. When property (5.16) holds, there exists an h0 > 0 such that, for all h ≤ h0,DFh(U∗) is an isomorphism of Y and the norm of its inverse is bounded independently ofh.

Proof: By writing the identity

DFh(U∗) = DF(U∗)(

Id−DF(U∗)−1(DF(U∗)−DFh(U∗)

)),

we observe that it suffices to check that

limh→0

‖DF(U∗)−DFh(U∗)‖E(Y) = 0.

We have

DF(U∗)−DFh(U∗) =(M(k∗)−Mh(k∗)

)DG(U∗) +

(DS(k∗)−DSh(k∗) 0

0 0

)G(U∗),

and we now prove successively that each term in the right-hand side tends to zero.1) Thanks to (5.18),it suffices to check that the mapping: W 7→ DG(U∗) ·W sends theunit sphere of Y into a bounded set of Y. We have, for W = (w, κ),

DG(U∗) ·W =(

(u∗ · ∇)w + (w · ∇)u∗

−2ν∗(·, k∗)∇u∗ : ∇w − (∂ξν∗(·, k∗)κ) |∇u∗|2).

When W runs though the unit sphere of Y, (u∗ · ∇)w+ (w · ∇)u∗ belongs to a boundedset of Lq(Ω)d for all q < 2 in dimension d = 2 and all q < 3

2 in dimension d = 3, andhence to a bounded set of Hτ (Ω)d for some τ , −1 < τ < 0. On the other hand, the twoterms in the second component of DG(U∗) ·W are bounded in Lq(Ω), for all q such that1q ≥ 1 − s

d and 1q ≥ 1 − 2s

d , respectively. So their sum belongs to a bounded set of Lq(Ω)with 1

q = 1 − sd , and it follows from the assumption on s that this space is imbedded in

Ht(Ω), t > d2 − 2. By combining this with (5.18), we obtain the convergence result.

2) It is readily checked that, for all F in H−1(Ω)d,(DS(k∗)κ

)F = S(k∗)

(−div (∂ξν∗(·, k∗)κ∇S(k∗)F )

),(

DSh(k∗)κ)F = Sh(k∗)

(−div (∂ξν∗(·, k∗)κ∇Sh(k∗)F )

),

(5.19)

19

so that(DS(k∗)−DSh(k∗) 0

0 0

)G(U∗) = −

((S(k∗)− Sh(k∗)

)div((∂ξν∗(·, k∗)κ)∇u∗

)0

)−(Sh(k∗) 0

0 0

)(div((∂ξν∗(·, k∗)κ)∇

(S(k∗)− Sh(k∗)

)((u∗ · ∇)u∗ − f

))0

).

There also, the convergence of the first term follows from the fact that (∂ξν∗(·, k∗)κ)∇u∗belongs to a compact set of L2(Ω)d×d, hence that div

((∂ξν∗(·, k∗)κ)∇u∗

)belongs to a

compact set of H−1(Ω)d when κ runs through the unit sphere of H1(Ω) ∩ L∞(Ω). Theconvergence of the second term is an easy consequence of (5.11) and the convergence of(S(k∗)− Sh(k∗)

)((u∗ · ∇)u∗ − f

)in H1(Ω)d.

The stability property (5.16) seems unsufficient in view of what follows. So we nowprove two modified results.

Lemma 5.3. The following result holds for any distribution G in H−1(Ω)

‖LhG‖L∞(Ω) ≤ c λh ‖G‖H−1(Ω), (5.20)

where the quantity λh is defined by

λh =

| log hmin|

12 if d = 2,

h− 1

2min if d = 3,

with hmin = minK∈Th

hK . (5.21)

Proof: Setting kh = LhG, we define the piecewise constant function kh by

∀K ∈ Th, kh|K =1

meas(K)

∫K

kh(x) dx,

and set: kh = kh − kh. We now estimate sucessively ‖kh‖L∞(Ω) and ‖kh‖L∞(Ω).1) We note that

‖kh‖L∞(Ω) = maxK∈Th

‖kh‖L∞(K).

By going to a reference element, we have on each K in Th and with the standard notation

‖kh‖L∞(K) = ‖ˆkh‖L∞(K) ≤ c ‖ˆkh‖H1(K),

where we have used the fact that all norms are equivalent on P2(K). Since kh has a

null integral on K, the same property holds for ˆkh on K, so that we derive from the

Bramble–Hilbert inequality (see for instance [6, Chap. I, Lemme 2.11])

‖kh‖L∞(K) ≤ c |ˆkh|H1(K),

and switching back to K yields

‖kh‖L∞(K) ≤ c h1− d2K |kh|H1(K).

20

Noting that kh − kh is constant on each K, so that |kh|H1(K) is equal to |kh|H1(K), andalso that the maximum of the |kh|H1(K) is bounded by |kh|H1(Ω), we obtain

‖kh‖L∞(Ω) ≤ c h1− d2min |kh|H1(Ω). (5.22)

2) Let p be a real number > 1 such that H1(Ω) is imbedded in Lp(Ω). Setting 1p + 1

p′ = 1,we have from the definition of kh and on each K

|kh| ≤meas(K)

1p′

meas(K)‖kh‖Lp(K) ≤ c h

− dpK ‖kh‖Lp(K).

This yields

‖kh‖L∞(Ω) ≤ c h− dpmin ‖kh‖Lp(Ω).

In dimension d = 2, we recall from [40] that the norm of the imbedding of H1(Ω) into

Lp(Ω) for any p < +∞ behaves like c√p and we take p = | log hmin|, so that

√p h− dpmin is

equal to e2 | log hmin|12 . In dimension d = 3 we take p equal to 6, such that h

− dpmin is equal

to h−12

min. This gives‖kh‖L∞(Ω) ≤ c λh ‖kh‖H1(Ω). (5.23)

Finally, it follows from (5.22) and (5.23) that

‖kh‖L∞(Ω) ≤ c λh ‖kh‖H1(Ω), (5.24)

and combining this result with (5.13) gives the desired estimate.

Lemma 5.4. The following result holds for any distribution G in L1(Ω)

‖LhG‖H1(Ω) ≤ c λh ‖G‖L1(Ω), ‖LhG‖L∞(Ω) ≤ c λ2h ‖G‖L1(Ω), (5.25)

where λh is defined in (5.21).

Proof: Owing to (5.24), it suffices to check the first part of (5.25). Setting once morekh = LhG, we observe from the definition (5.9) of Lh and a Poincare–Friedrichs inequalitythat

‖kh‖2H1(Ω) ≤c

α‖G‖L1(Ω)‖kh‖L∞(Ω).

Using once more (5.24) yields the desired estimate.

Lemma 5.5. If the function ν belongs to W 2,∞(R), the mapping DFh satisfies the fol-lowing Lipschitz property, for all V1 and V2 in a bounded subset of Y,

‖DFh(V1)−DFh(V2)‖E(Y) ≤ c λ2h ‖V1 − V2‖Y , (5.26)

where λh is defined in (5.21).

Proof: Setting Vi = (vi, `i), we have

DFh(V1)−DFh(V2) =(Mh(`1)−Mh(`2)

)DG(V1) +Mh(`2)

(DG(V1)−DG(V2)

)+(DMh(`1)−DMh(`2)

)G(V1) +DMh(`2)

(G(V1)− G(V2)

).

21

Let W = (w, κ) be any function in Y. The proof now proceeds in two steps. All constantsc in what follows only depend on the norms ‖V1‖Y , ‖V2‖Y and ‖ν‖W 2,∞(R).1) We have(Mh(`1)−Mh(`2)

)DG(V1)W

=Mh(`1)(−div

((ν∗(·, `1)− ν∗(·, `2))∇Sh(`2)

((v1 · ∇)w + (w · ∇)v1

))0

),

which yields in an obvious way

‖(Mh(`1)−Mh(`2)

)DG(V1)W‖Y ≤ c ‖V1 − V2‖Y‖W‖Y . (5.27)

On the other hand, since ∂ξν∗(x, ·) is zero on Ω` while on Ωt it coincides with ∂ξν, itbelongs to W 1,∞(R) for a.e. x in Ω. So, combining the second part of (5.19) with afurther triangle inequality also yields

‖((DMh(`1)−DMh(`2)

)κ)G(V1)‖Y ≤ c ‖V1 − V2‖Y‖W‖Y . (5.28)

2) It follows from (5.19) that

Mh(`2)(DG(V1)−DG(V2)

)+DMh(`2)

(G(V1)− G(V2)

)=Mh(`2)

(ΓSΓL

), (5.29)

where

〈ΓS ,W 〉 =((v1 − v2) · ∇

)w + (w · ∇)(v1 − v2)

− div((∂ξν∗(·, `2)κ

)∇S∗h(`2)

((v1 · ∇)v1 − (v2 · ∇)v2

)),

〈ΓL,W 〉 = −2ν∗(·, `1)∇v1 : ∇w + 2ν∗(·, `2)∇v2 : ∇w− ∂ξν∗(·, `1)κ |∇v1|2 + ∂ξν

∗(·, `2)κ |∇v2|2.

To conclude, we note that:• It can be checked through technical arguments that

‖〈ΓS ,W 〉‖H−1(Ω)d ≤ c ‖V1 − V2‖Y‖W‖Y ,

This estimate together with (5.11) gives the Lipschitz–continuity of the first component in(5.29).• Similar arguments also yield

‖〈ΓL,W 〉‖L1(Ω) ≤ c ‖V1 − V2‖Y‖W‖Y .

Together with the first part of (5.25) this implies the Lipschitz–continuity of the secondcomponent in (5.29) with values in H1

0 (Ω), with Lipschitz constant smaller than c λh.• By combining the previous estimate with the second part of (5.25), we derive that thesecond component in (5.29) is also Lipschitz–continuous with values in L∞(Ω), but nowwith Lipschitz constant smaller than c λ2

h.Combining these last results with (5.27) and (5.28) leads to the desired property.

22

The next lemma relies on estimates (5.12), (5.14) and (5.15). We also derive thefollowing result by combining (5.15) and (5.16), using the imbedding of Hs+sd(Ω) intoW s,∞(Ω) for a any real number sd > d

2 and applying an interpolation argument: For anydata G in H−1(Ω)d such that LG belongs to Hs+sd(Ω) for 1 ≤ s ≤ 3,

‖(L − Lh)G‖L∞(Ω) ≤ c hs ‖LG‖Hs+sd (Ω). (5.30)

Lemma 5.6. Let (u∗, p∗, k∗) be a solution of problem (4.4) which belongs to Hs+1(Ω)d ×Hs(Ω)×Hs+sd(Ω), 0 < s ≤ 2, for a fixed real number sd > d

2 . When property (5.16) holds,the following estimate is satisfied

‖Fh(U∗)‖Y ≤ c(u∗, p∗, k∗)hs, (5.31)where the constant c(u∗, p∗, k∗) only depends on the solution (u∗, p∗, k∗).

Proof: We have‖Fh(U∗)‖Y = ‖F(U∗)−Fh(U∗)‖Y = ‖

(M(k∗)−Mh(k∗)

)G(U∗)‖Y ,

so that the lemma is a direct consequence of (5.12), (5.14) and (5.30) together with theregularity properties of U∗ = −M(k∗)G(U∗).

Thanks to Lemmas 5.2, 5.5 and 5.6, we are in a position to apply the Brezzi–Rappaz–Raviart theorem [9]. We also recall the existence of a discrete inf-sup condition betweenthe spaces Xh and Mh, see [20, Chap. II, Cor. 4.1]: There exists a constant β > 0independent of h such that

∀qh ∈Mh, supvh∈Xh

−∫

Ω(div vh)(x)qh(x) dx‖vh‖H1(Ω)d

≥ β ‖qh‖L2(Ω). (5.32)

This leads to the main result of this section.

Theorem 5.7. Let (u∗, p∗, k∗) be a solution of problem (4.4) which belongs to Hs+1(Ω)d×Hs(Ω)×Hs+sd(Ω), d−2 < s ≤ 2, for a fixed real number sd > d

2 , and such that DF∗(U∗),with U∗ = (u∗, k∗)T, is an isomorphism of Y. We moreover assume that property (5.16)holds, that the function ν belongs to W 2,∞(R) and that

limh→0

λ2h h

s = 0, (5.33)

where λh is defined in (5.21). Then, there exist positive numbers κ and h0 such that, forany h ≤ h0, problem (5.3) has a unique solution (uh, ph, kh) such that (uh, kh) belongs tothe ball of Y with centre (u∗, k∗) and radius κλ−2

h . Moreover this solution satisfies‖u∗ − uh‖H1(Ω)d + ‖p∗ − ph‖L2(Ω) + ‖k∗ − kh‖H1(Ω)∩L∞(Ω) ≤ c(u∗, p∗, k∗)hs, (5.34)

where the constant c(u∗, p∗, k∗) only depends on the solution (u∗, p∗, k∗).

Estimate (5.34) is fully optimal since it leads to a convergence order equal to 2 forvery smooth solutions (u∗, p∗, k∗). Moreover assumption (5.33) is not at all restrictivein dimension d = 2. But neither this assumption nor the regularity property requiredon the solution (u∗, p∗, k∗) is likely in dimension d = 3. However the convergence of thediscretization when the following condition holds

limh→0

h−1min h

2 = 0, (5.35)

can easily be derived from Theorem 5.7. This last condition is now much more likely andcan be enforced in the adaptivity process.

23

6. A posteriori estimates.

For each K in Th, we introduce the set EK of edges (d = 2) or faces (d = 3) of Kwhich are not contained in ∂Ω. The diameters (or lengths) of K and of each e in EK aredenoted by hK and he, respectively. For each e in EK , [·]e stands for the jump through ein a fixed direction (it seems useless to make the sign precise). We also consider the space

Zh =vh ∈ L2(Ω)d; ∀K ∈ Th, vh|K ∈ P0(K)d

, (6.1)

and, assuming that the datum f belongs to L2(Ω)d, we choose an approximation fh of fin Zh.

We again use the real number ρ defined in (4.14) and we set

1ρ

+1ρ′

= 1. (6.2)

We are now in a position to define the two types of error indicators which are needed forapplying the algorithm described in Section 2.(i) An indicator linked to the modelization errorThe definition of this indicator relies on Proposition 4.5. The error indicator ηm is givenby

ηm = ‖div(µ∗(·, kh)∇uh

)‖W−1,ρ(Ω)d + ‖µ∗(·, kh) |∇uh|2‖W−1,ρ(Ω), (6.3)

where the function µ∗ is defined in (4.16).(ii) Indicators linked to the finite element errorThese indicators are of residual type and are now standard both for the Stokes and Laplaceequations, see [41, §1.2]. For each K in Th, the error indicator ηK is given by

ηK = ηSK + ηLK , (6.4)

with

ηSK = hK ‖fh + div(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph‖Lρ(K)d

+∑e∈EK

h1ρe ‖[ν∗(·, kh) ∂nuh]e‖Lρ(e)d + ‖divuh‖Lρ(K),

ηLK = hK ‖ν∗(·, kh) |∇uh|2 + α∆kh‖Lρ(K) +∑e∈EK

h1ρe ‖[α∂nkh]e‖Lρ(e).

(6.5)

It must be noted that each term which appears in the definition of the indicators ηK iseasy to compute once the discrete solution (uh, ph, kh) is known.

In a first step, we intend to prove an upper bound for the error between a solution(u, p, k) of problem (3.3) and the solution (uh, ph, kh) of problem (5.3) in a neighbourhoodof it (see Proposition 4.5 and Theorem 5.7), in the norm of the space W 1,ρ

0 (Ω)d×Lρ(Ω)×W 1,ρ

0 (Ω). Relying on the triangle inequality

‖u− uh‖W 1,ρ(Ω)d ≤ ‖u− u∗‖W 1,ρ(Ω)d + ‖u∗ − uh‖W 1,ρ(Ω)d ,

24

and similar ones for the other unknowns, we proceed in two steps. We again use the spaceX defined in (4.8).

Proposition 6.1. If the function ν belongs to W 2,∞(R) and Assumption 4.4 is satisfied,let (u, p, k) be a solution of problem (3.3) such that DF(U) with U = (u, k)T, is anisomorphism of X . There exists a positive number R only depending on this solution suchthat the following a posteriori error estimate holds

‖u− u∗‖W 1,ρ(Ω)d + ‖k − k∗‖W 1,ρ(Ω)

≤ c ηm + c(f)(‖u∗ − uh‖W 1,ρ(Ω`)d + ‖k∗ − kh‖W 1,ρ(Ω`)

) (6.6)

for any solution (u∗, p∗, k∗) of problem (4.4) such that (u∗, k∗) belongs to the ball of Xwith centre (u, k) and radius R and for a constant c(f) only depending on the data f .

Proof: It follows from Assumption 4.4 and Proposition 4.3 that the solution (u∗, p∗, k∗)satisfies, for a real number q1 > ρ,

ν0 ‖u∗‖W 1,q1 (Ω)d + α ‖k∗‖W 1,q1 (Ω) ≤ c(‖f‖W−1,q1 (Ω)d + ‖f‖2H−1(Ω)d

). (6.7)

We also need the inequality (which is easily derived by a duality argument)

∀w ∈ Lρ(Ω)d, ‖divw‖W−1,ρ(Ω) ≤ ‖w‖Lρ(Ω)d . (6.8)

Moreover, noting that all assumptions of Proposition 4.5 are satisfied and that the supportof µ∗ is contained in Ω`, we derive from (4.17) combined with (6.8) that

‖u− u∗‖W 1,ρ(Ω)d + ‖k − k∗‖W 1,ρ(Ω)d

≤ c(ηm + ‖µ(k∗)∇u∗ − µ(kh)∇uh‖Lρ(Ω`)d×d

+ ‖µ∗(·, k∗) |∇u∗|2 − µ∗(·, kh) |∇uh|2‖W−1,ρ(Ω)

).

(6.9)

We then use the further triangle inequality

‖µ(k∗)∇u∗ − µ(kh)∇uh‖Lρ(Ω`)d×d ≤ ‖(µ(k∗)− µ(kh)

)∇u∗‖Lρ(Ω`)d×d

+ ‖µ(kh)∇(u∗ − uh)‖Lρ(Ω`)d×d .

The second term in the right hand-side is obviously bounded by

‖µ(kh)∇(u∗ − uh)‖Lρ(Ω`)d×d ≤ (ν2 − ν0) ‖u∗ − uh‖W 1,ρ(Ω)d .

To evaluate the first term, we consider the four cases where• ν1

√(k∗)ε+ ≥ ν2 − ν0 and ν1

√(kh)ε+ ≥ ν2 − ν0,

• ν1

√(k∗)ε+ < ν2 − ν0 and ν1

√(kh)ε+ ≥ ν2 − ν0,

• ν1

√(k∗)ε+ ≥ ν2 − ν0 and ν1

√(kh)ε+ < ν2 − ν0,

• ν1

√(k∗)ε+ < ν2 − ν0 and ν1

√(kh)ε+ < ν2 − ν0,

and observe that, in all of them,

|µ(k∗)− µ(kh)| ≤ ν1

∣∣√(k∗)ε −√

(kh)ε∣∣.

25

Since q1 is > ρ, setting 1r = 1

ρ −1q1

and noting that W 1,ρ(Ω`) is imbedded in Lr(Ω`), weobtain

‖(µ(k∗)− µ(kh)

)∇u∗‖Lρ(Ω`)d×d ≤ ν1‖

√(k∗)ε −

√(kh)ε‖Lr(Ω`)‖∇u

∗‖Lq1 (Ω)d×d

≤ ν1

2√ε‖k∗ − kh‖W 1,ρ(Ω`)‖u

∗‖W 1,q1 (Ω)d ,

where the last estimate is obtained by multiplying and dividing by√

(k∗)ε +√

(kh)ε.Combining all this with (6.7) gives

‖µ(k∗)∇u∗−µ(kh)∇uh‖Lρ(Ω`)d×d ≤ c(f)(‖u∗−uh‖W 1,ρ(Ω`)d+‖k∗−kh‖W 1,ρ(Ω`)

), (6.10)

where the quantity c(f) only depends on f . On the other hand, we use the inequality

‖µ∗(·, k∗) |∇u∗|2 − µ∗(·, kh) |∇uh|2‖W−1,ρ(Ω)

≤ ‖(µ∗(·, k∗)− µ∗(·, kh)

)|∇u∗|2‖W−1,ρ(Ω)

+ ‖µ∗(·, kh)∇(u∗ + uh) · ∇(u∗ − uh)‖W−1,ρ(Ω),

whence, by combining the same arguments as previously with the imbedding of Lr(Ω) intoW−1,ρ(Ω) for 1

r = 1ρ + 1

d and again using (6.7),

‖µ∗(·, k∗) |∇u∗|2 − µ∗(·, kh) |∇uh|2‖W−1,ρ(Ω)

≤ c(f)(‖u∗ − uh‖W 1,ρ(Ω`)d + ‖k∗ − kh‖W 1,ρ(Ω`)

).

(6.11)

Combining (6.10) and (6.11) with (6.9) leads to the desired estimate.

To prove the analogous estimate for the pressure, we need the following inf-sup con-dition which is proven in [2, Cor. 3.2] for instance: There exists a constant βρ such that

∀q ∈ Lρ(Ω), supv∈W 1,ρ′

0 (Ω)d

−∫

Ω(div v)(x)q(x) dx‖v‖W 1,ρ′ (Ω)d

≥ βρ ‖q‖Lρ(Ω). (6.12)

Note that this condition holds with ρ replaced by any r, 1 < r < +∞. Indeed, we observefrom (3.3) and (4.4) that, for any function v in H1

0 (Ω)d,

−∫

Ω

(div v)(x)(p− p∗)(x) dx = −∫

Ω

(ν(k)∇u− ν∗(k∗)∇u∗

): ∇v dx

−∫

Ω

((u · ∇)u− (u∗ · ∇)u∗

)· v dx.

Thus, using the density of H10 (Ω) into W 1,ρ′

0 (Ω), we derive from the inf-sup condition (6.12)an estimate for ‖p− p∗‖Lρ(Ω) in terms of the already evaluated quantities.

Corollary 6.2. If the assumptions of Proposition 6.1 hold and for the R introduced inthis proposition, the following a posteriori error estimate holds

‖p− p∗‖Lρ(Ω) ≤ c ηm + c(f)(‖u∗ − uh‖W 1,ρ(Ω`)d + ‖k∗ − kh‖W 1,ρ(Ω`)

), (6.13)

26

for any solution (u∗, p∗, k∗) of problem (4.4) such that (u∗, k∗) belongs to the ball of Xwith centre (u, k) and radius R and for a constant c(f) only depending on the data f .

Proving the bound for the second part of the error still relies on the theorem ofJ. Pousin and J. Rappaz [38]. We first observe that the triple (uh, ph, kh) satisfies thesystem of residual equations: For all v in W 1,ρ′

0 (Ω)d and vh in Xh,∫Ω

ν∗(x, kh)∇uh : ∇v dx+∫

Ω

(uh · ∇)uh · v dx−∫

Ω

(div v)(x)ph(x) dx− 〈f ,v〉

= −∑K∈Th

(∫K

(f + div

(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph

)(x) · (v − vh)(x) dx

+12

∑e∈EK

∫e

[ν∗(·, kh) ∂nuh]e(τ ) · (v − vh)(τ ) dτ),

(6.14)and, for all χ in W 1,ρ′

0 (Ω) and χh in Yh,

α

∫Ω

grad kh · gradχdx−∫

Ω

ν∗(x, kh) |∇uh|2(x)χ(x) dx

= −∑K∈Th

(∫K

(ν∗(·, kh) |∇uh|2 + α∆kh

)(x) (χ− χh)(x) dx

+12

∑e∈EK

∫e

α [∂nkh]e(τ ) (χ− χh)(τ ) dτ).

(6.15)

However, the discrete velocity uh is no longer divergence-free in the general case.So we derive from the inf-sup condition (6.12), with ρ replaced by ρ′, the existence of afunction u in W 1,ρ

0 (Ω)d such that

divu = divuh in Ω and ‖u‖W 1,ρ(Ω)d ≤ βρ′ ‖divuh‖Lρ(Ω). (6.16)

Thus, when setting u0 = uh −u, equation (6.14) can equivalently be written: For all v inW 1,ρ′

0 (Ω)d, vh in Xh and q in L20(Ω),∫

Ω

ν∗(x, kh)∇u0 : ∇v dx+∫

Ω

(u0 · ∇)u0 · v dx−∫

Ω

(div v)(x)ph(x) dx− 〈f ,v〉

= −∑K∈Th

(∫K

(f + div

(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph

)(x) · (v − vh)(x) dx

+12

∑e∈EK

∫e

[ν∗(·, kh) ∂nuh]e(τ ) · (v − vh)(τ ) dτ)− r(u,v),

−∫

Ω

(divu0)(x)q(x) dx = 0,

(6.17)where the quantity r(u, ·) is defined by

r(u,v) =∫

Ω

ν∗(x, kh)∇u : ∇v dx+∫

Ω

(u · ∇)u · v dx

+∫

Ω

(u · ∇)u0 · v dx+∫

Ω

(u0 · ∇)u · v dx.(6.18)

27

When using the notation introduced in Section 4, we write the system (6.17)− (6.15)in the more compact form

U0 +(S 00 L

)G∗(U0) =

(S 00 L

)R(U0), (6.19)

where U0 stands for the pair (u0, kh) and the two components RS(U0) and RL(U0) ofR(U0) are defined by duality

∀v ∈W 1,ρ′

0 (Ω)d, 〈RS(U0),v〉

= −∑K∈Th

(∫K

(f + div

(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph

)(x) · (v −Πhv)(x) dx

+12

∑e∈EK

∫e

[ν∗(·, kh) ∂nuh]e(τ ) · (v −Πhv)(τ ) dτ)− r(u,v),

∀χ ∈W 1,ρ′

0 (Ω), 〈RL(U0), χ〉 = −∑K∈Th

(∫K

(ν(kh) |∇uh|2 + α∆kh

)(x) (χ−Πhχ)(x) dx

+12

∑e∈EK

∫e

α [∂nkh]e(τ ) (χ−Πhχ)(τ ) dτ),

where the operator Πh is defined below.

As standard in a posteriori analysis, the next result requires that Πh is a Clementtype operator, see [13], more precisely an operator from L1(Ω) with values in Yh and suchthat, for any function ϕ in W 1,ρ′

0 (Ω), for all K in Th and e in EK ,

‖ϕ−Πhϕ‖Lρ′ (K) ≤ c hK ‖ϕ‖W 1,ρ′ (∆K),

‖ϕ−Πhϕ‖Lρ′ (e) ≤ c h1− 1

ρ′

K ‖ϕ‖W 1,ρ′ (∆e),

(6.20)

where ∆K and ∆e denote the union of the elements of Th that share at least a vertex withK and e, respectively. Such an operator was first constructed in [13], where the first part ofestimate (6.20) is proved for ρ = ρ′ = 2 in the two-dimensional case. A modified operatoris constructed in [6, Th. IX.3.11], where the first part of estimate (6.20) is extended tothe case d = 3 and to any value of ρ. The second part of (6.20) is proved for this sameoperator in [6, Cor. IX.3.12]. We are now in a position to prove the next statement.

Proposition 6.3. Assume that there exists a constant R] such that all solutions of problem(5.3) satisfy

‖divuh‖Lρ(Ω) ≤ R]. (6.21)

If the function ν belongs to W 2,∞(R), let (u∗, p∗, k∗) be a solution of problem (4.4) whichbelongs to X and such that DF∗(U∗) with U∗ = (u∗, k∗)T, is an isomorphism of X . Then,there exists a positive number R[ only depending on this solution such that the following aposteriori error estimate holds

‖u∗ − uh‖W 1,ρ(Ω)d + ‖k∗ − kh‖W 1,ρ(Ω) ≤ c( ∑K∈Th

(ηρK + hρK ‖f − fh‖

ρLρ(K)d

)) 1ρ

, (6.22)

28

for a solution (uh, ph, kh) of problem (5.3) such that (uh, kh) belongs to the ball of X withcentre (u∗, k∗) and radius R[.

Proof: We first note from (6.16) that

‖u‖W 1,ρ(Ω)d ≤( ∑K∈Th

(ηSK)ρ

) 1ρ , (6.23)

and we use the triangle inequality

‖u∗ − uh‖W 1,ρ(Ω)d ≤ ‖u∗ − u0‖W 1,ρ(Ω)d + ‖u‖W 1,ρ(Ω)d .

Indeed, the same arguments as in Section 4 yield that all the assumptions of [41, Prop.2.1] are satisfied by the solution U∗ of problem (4.11), so that applying this propositionleads to the following result: There exists a constant R0 > 0 independent of h such thatany solution U0 of (6.19) in the ball with centre U∗ and radius R0 satisfies

‖U∗ − U0‖X∗ ≤ c∥∥∥∥(S 0

0 L

)R(U0)

∥∥∥∥X∗.

The pair (uh = u0 + u, kh) belongs to the ball with centre U∗ and radius R[ = R0 + R].On the other hand, the standard properties of the operators S and L give

‖u∗ − u0‖W 1,ρ(Ω)d + ‖k∗ − kh‖W 1,ρ(Ω)

≤ c (‖RS(U0)‖W−1,ρ(Ω)d + ‖RL(U0)‖W−1,ρ(Ω)

).

(6.24)

We now evaluate successively each term in the right-hand side.1) By combining (6.16), (6.21) and the fact that ‖u0‖W 1,ρ(Ω)d is bounded, we easily derivethat

|r(u,v)| ≤ c ‖divuh‖Lρ(Ω) ‖v‖W 1,ρ′ (Ω)d .

To bound the other terms in RS(U0), we use estimates (6.20), a Holder inequality, thefact that each element of Th is included in a finite number of ∆K and ∆e, only dependingon the regularity parameter σ, and a further triangle inequality to replace f by fh. Allthis leads to

|〈RS(U0),v〉| ≤ c( ∑K∈Th

((ηSK)ρ + hρK ‖f − fh‖

ρLρ(K)d

)) 1ρ ‖v‖W 1,ρ′ (Ω)d . (6.25)

2) Similarly, using (6.20) and the same arguments as previously gives

|〈RL(U0), χ〉| ≤ c( ∑K∈Th

(ηLK)ρ) 1ρ ‖χ‖W 1,ρ′ (Ω). (6.26)

Inserting (6.25) and (6.26) into (6.24) and combining this with (6.23) yield the desiredestimate.

Remark 6.4. Assumption (6.21) can easily be avoided by replacing the Stokes operatorS by the “complete” Stokes operator which associates with any data (F , G) in H−1(Ω)d×

29

L20(Ω), the solution (u, p) of a generalized problem (4.6) (see [41, §3.5] for more details).

However we prefer to work with the operator S in order that the assumptions of Proposition6.3 are coherent with their analogues in Proposition 6.1. Anyhow assumption (6.21) is notrestrictive since it results from the convergence of the discretization.

Proving the estimate on the pressure now relies on the following argument. By com-bining (6.14) and the first equation of problem (4.4), we obtain for all v in W 1,ρ′

0 (Ω)d andvh in Xh,

−∫

Ω

(div v)(x)(p∗ − ph)(x) dx = −∫

Ω

ν∗(x, k∗)∇u∗ : ∇v dx

+∫

Ω

ν∗(x, kh)∇uh : ∇v dx−∫

Ω

(u∗ · ∇)u∗ · v dx+∫

Ω

(uh · ∇)uh · v dx

+∑K∈Th

(∫K

(f + div

(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph

)(x) · (v − vh)(x) dx

− 12

∑e∈EK

∫e

[ν∗(·, kh) ∂nuh]e(τ ) · (v − vh)(τ ) dτ).

(6.27)Thus, the inf-sup condition (6.12) leads to a bound for ‖p−ph‖Lρ(Ω) in terms of the ηK andthe quantities ‖u∗−uh‖W 1,ρ(Ω)d and ‖k∗− kh‖W 1,ρ(Ω) which are evaluated in Proposition6.3.

Corollary 6.5. If the assumptions of Proposition 6.3 hold and for the R[ introduced inthis proposition, the following a posteriori error estimate holds

‖p∗ − ph‖Lρ(Ω) ≤ c( ∑K∈Th

(ηρK + hρK ‖f − fh‖

ρLρ(K)d

)) 1ρ

, (6.28)

for a solution (uh, ph, kh) of problem (5.3) such that (uh, kh) belongs to the ball of X ∗ withcentre (u∗, k∗) and radius R[.

We are now interested in upper bounds for the error indicators.

Proposition 6.6. If Assumption 4.4 is satisfied, the following estimate holds for theindicator ηm defined in (6.3),

ηm ≤ c(f)(‖u− u∗‖W 1,ρ(Ω)d + ‖p− p∗‖Lρ(Ω) + ‖k − k∗‖W 1,ρ(Ω)

)+ c′(f)

(‖u∗ − uh‖W 1,ρ(Ω`)d + ‖k∗ − kh‖W 1,ρ(Ω`)

),

(6.29)

for constants c(f) and c′(f) only depending on the data f .

Proof: It is performed in three steps.1) By subtracting the first line of problem (4.4) from the first line of problem (3.3), weobtain for any v in W 1,ρ′

0 (Ω)d and with obvious notation for the duality pairing∫Ω

(ν(k)∇u− ν(k∗)∇u∗

): ∇v dx+

∫Ω

((u · ∇)u− (u∗ · ∇)u∗

)· v(x) dx

−∫

Ω

(div v)(x)(p− p∗)(x) dx =⟨

div(µ∗(·, k∗)∇u∗

),v⟩.

30

Thus, we obtain

‖div(µ∗(·, k∗)∇u∗

)‖W−1,ρ(Ω)d ≤ ‖ν(k)∇u− ν(k∗)∇u∗‖Lρ(Ω)d×d

+ ‖(u · ∇)u− (u∗ · ∇)u∗‖Lρ(Ω)d + ‖p− p∗‖Lρ(Ω).

It follows from standard arguments and estimate (6.7) that

‖div(µ∗(·, k∗)∇u∗

)‖W−1,ρ(Ω)d

≤ c(f)(‖u− u∗‖W 1,ρ(Ω)d + ‖p− p∗‖Lρ(Ω) + ‖k − k∗‖W 1,ρ(Ω)

).

(6.30)

2) Similarly, by subtracting the third line of problem (4.3) from the third line of problem(1.2), we obtain

−α∆(k − k∗) = ν(k) |∇u|2 − ν(k∗) |∇u∗|2 − µ∗(·, k∗) |∇u∗|2,

whence

‖µ∗(·, k∗) |∇u∗|2‖W−1,ρ(Ω) ≤ c(‖k − k∗‖W 1,ρ(Ω)d + ‖ν(k) |∇u|2 − ν(k∗) |∇u∗|2‖W−1,ρ(Ω)

).

Owing to (6.7), this yields

‖µ∗(·, k∗) |∇u∗|2‖W−1,ρ(Ω) ≤ c(f)(‖u− u∗‖W 1,ρ(Ω)d + ‖k − k∗‖W 1,ρ(Ω)

). (6.31)

3) A triangle inequality and (6.8) lead to

ηm ≤ ‖div(µ∗(·, k∗)∇u∗

)‖W−1,ρ(Ω)d + ‖µ∗(·, k∗) |∇u∗|2‖W−1,ρ(Ω)

+ ‖µ(k∗)∇u∗ − µ(kh)∇uh‖Lρ(Ω`)d×d

+ ‖µ∗(·, k∗) |∇u∗|2 − µ∗(·, kh) |∇uh|2‖W−1,ρ(Ω).

We conclude by using (6.10) and (6.11), combined with (6.30) and (6.31).

Proving the upper bounds for the indicators ηK presents two difficulties:• the quantities that appear in their definition are no longer polynomial, due to the nonpolynomial function ν∗,• the norms in their definition are no longer Hilbertian.To handle both of them, we use some arguments presented in [41, §3.3] and [41, Lemma3.3].

We bound separately the terms ηSK and ηLK . We recall from Remark 5.1 the definitionof the function νh: For each K in Th, νh|K belongs to P2(K) and is equal to ν∗(a, kh(a))at each vertex or midpoint a of an edge of K. According to [41, §3.3], we introduce thequantities

εSK = hK ‖div((ν∗(·, kh)− νh(·)

)∇uh

)‖Lρ(K)d

+∑e∈EK

h1ρe ‖[

(ν∗(·, kh)− νh(·)

)∂nuh]e‖Lρ(e)d .

(6.32)

31

In a first step, we prove a bound for the quantity εSK . For each K in Th, we denote by ωKthe union of elements of Th that share at least an edge (d = 2) or a face (d = 3) with K.

Lemma 6.7. The following estimate holds for all K in Th,

εSK ≤ c hK ‖grad kh‖L∞(K)d ‖∇uh‖Lρ(ωK)d×d . (6.33)

Proof: We bound successively the two terms in εSK .1) We have

div((ν∗(·, kh)− νh(·)

)∇uh

)=(ν∗(·, kh)− νh(·)

)∆uh + grad

(ν∗(·, kh)− νh(·)

): ∇uh,

whence

hK ‖div((ν∗(·, kh)− νh(·)

)∇uh

)‖Lρ(K)d

≤ hK ‖(ν∗(·, kh)− νh(·)

)‖L∞(K) ‖∆uh‖Lρ(K)d

+ hK ‖grad(ν∗(·, kh)− νh(·)

)‖L∞(K)d ‖∇uh‖Lρ(K)d×d

.

Noting that νh(·) is the Lagrange interpolate of ν∗(·, kh), therefore we derive from standardestimates [6, Lemmes IX.1.1 & IX.1.2] and the fact that ν∗(·, kh) belongs to W 1,∞(Ω)

‖(ν∗(·, kh)−νh(·)

)‖L∞(K) +hK ‖grad

(ν∗(·, kh)−νh(·)

)‖L∞(K)d ≤ c hK ‖grad kh‖L∞(K)d .

Combining this with a standard inverse inequality [6, Prop. VII.4.1] to bound the term‖∆uh‖Ld(K)d , we obtain

hK ‖div((ν∗(·, kh)−νh(·)

)∇uh

)‖Lρ(K)d ≤ c hK ‖grad kh‖L∞(K)d ‖∇uh‖Lρ(K)d×d . (6.34)

2) Similarly, we have

h1ρe ‖[

(ν∗(·, kh)− νh(·)

)∂nuh]e‖Lρ(e)d ≤ h

1ρe ‖ν∗(·, kh)− νh‖L∞(e)‖[∂nuh]e‖Lρ(e)d .

Using the properties of the Lagrange interpolation operator on e together with the factthat the trace operator from W 1,∞(K) into W 1,∞(e) has a norm bounded independentlyof hK , we derive

‖ν∗(·, kh)− νh‖L∞(e) ≤ c he ‖grad kh‖L∞(K)d .

On the other hand, combining a duality argument together with a less standard inverseinequality leads to

‖[∂nuh]e‖Lρ(e)d ≤ c h− 1ρ

e ‖[∂nuh]e‖W− 1ρ,ρ

(e)d≤ c′ h−

1ρ

e ‖∇uh‖Lρ(ωK)d×d .

All this yields

h1ρe ‖[

(ν∗(·, kh)− νh(·)

)∂nuh]e‖Lρ(e)d ≤ c he ‖grad kh‖L∞(K)d ‖∇uh‖Lρ(ωK)d×d . (6.35)

The desired estimate follows from (6.34) and (6.35) by noting that he ≤ hK .

32

To derive the error for the ηSK , we write the residual equation (6.14) in the modifiedform: For all v in W 1,ρ′

0 (Ω)d,∫Ω

(ν∗(x, k∗)∇u∗ − ν∗(x, kh)∇uh

): ∇v dx

+∫

Ω

((u∗ · ∇)u∗ − (uh · ∇)uh

)· v dx−

∫Ω

(div v)(x)(p∗ − ph)(x) dx

=∑K∈Th

(∫K

(fh + div

(νh(·)∇uh

)− (uh · ∇)uh − grad ph

)(x) · v(x) dx

+∫K

(f − fh)(x) · v(x) dx+∫K

div((ν∗(·, kh)− νh(·)

)(x)∇uh

)· v(x) dx

+12

∑e∈EK

(∫e

[νh(·) ∂nuh]e(τ ) · v(τ ) dτ

+∫e

[(ν∗(·, kh)− νh(·)

)∂nuh]e(τ ) · v(τ ) dτ

)).

(6.36)We also have, for all q in L2

0(Ω),∫Ω

(div (u− uh)

)(x)q(x) dx = −

∫Ω

(divuh

)(x)q(x) dx. (6.37)

Even if equation (6.36) is rather complex, the estimate for ηSK is now derived from theseequations by standard arguments, that we present in an abridged way.

Proposition 6.8. If Assumption 4.4 is satisfied, the following estimate holds for theindicators ηSK defined in (6.5), K ∈ Th,

ηSK ≤ c(f)(‖u∗ − uh‖W 1,ρ(ωK)d + ‖p∗ − ph‖Lρ(ωK) + ‖k∗ − kh‖W 1,ρ(ωK)

)+ c

(hK ‖f − fh‖Lρ(ωK)d +

∑κ⊂ωK

εSκ), (6.38)

for a constant c(f) only depending on the data f .

Proof: We evaluate separately the three terms in ηSK .1) Setting

ϕh = fh + div(νh(·)∇uh

)− (uh · ∇)uh − grad ph,

and using [41, Lemma 3.3], we derive that, for a finite-dimensional subspace VK of H1(K)d

on the reference triangle or tetrahedron K, denoting by VK the space of functions on Kobtained from VK by affine transformation,

‖ϕh‖Lρ(K)d ≤ c supw∈VK

∫Kϕh(x)ψK(x) · w(x) dx‖w‖Lρ′ (K)d

,

where ψK denotes the bubble function on K, equal to the product of the barycentriccoordinates associated with the vertices of K. Thus, taking v equal to ψKw in (6.36),where w runs through VK and using the standard inverse inequality [6, Prop. VII.4.1]

‖wψK‖W 1,ρ′ (K)d ≤ c h−1K ‖wψK‖Lρ′ (K)d ≤ c h

−1K ‖w‖Lρ′ (K)d ,

33

we obtain the desired bound for hK ‖ϕh‖Lρ(K)d . A further triangle inequality involvingεSK leads to the bound for

hK ‖fh + div(ν∗(·, kh)∇uh

)− (uh · ∇)uh − grad ph‖Lρ(K)d .

2) Setting ϕh = [νh(·) ∂nuh]e, we start from the similar formula

‖ϕh‖Lρ(e)d ≤ c supw∈Ve

∫eϕh(x)ψe(x) · w(x) dx

‖w‖Lρ′ (e)d,

where Ve is contructed by affine transformation from a finite-dimensional space Ve on thereference edge or face e, and ψe is the bubble function on e. Next, assuming that e is anedge of another element K ′, we set

v =Re,κ

(ψew) on κ ∈ K,K ′,

0 on Ω \ (K ∪K ′),

where Re,κ denotes a lifting operator from functions on e vanishing on ∂e into functionson κ vanishing on ∂κ\ e, constructed by affine transformation from a fixed lifting operator

from e onto K. This leads to the bound for h1ρe ‖[νh(·) ∂nuh]e‖Lρ(e)d . A further triangle

inequality involving εSK finally gives the bound for h1ρe ‖[ν∗(·, kh) ∂nuh]e‖Lρ(e)d .

3) It follows from (6.37) in an obvious way that

‖divuh‖Lρ(K) ≤ c |u∗ − uh|W 1,ρ(K)d .

This concludes the proof.

The arguments for bounding ηLK are similar but simpler. So we skip the proofs. Wenext introduce the quantities

εLK = hK ‖(ν∗(·, kh)− νh(·)

)|∇uh|2‖Lρ(K). (6.39)

Lemma 6.9. The following estimate holds for all K in Th,

εLK ≤ c h2K ‖grad kh‖L∞(K)d ‖∇uh‖2L2ρ(K)d×d . (6.40)

We need the following modified form of the residual equation (6.15): For all χ inW 1,ρ′

0 (Ω),

α

∫Ω

grad (k∗ − kh) · gradχdx

−∫

Ω

(ν∗(x, k∗) |∇u∗|2 − ν∗(x, kh) |∇uh|2

)(x)χ(x) dx

=∑K∈Th

(∫K

(νh(x) |∇uh|2(x) + α∆kh

)(x)χ(x) dx

−∫K

(ν∗(x, kh)− νh(x)

)|∇uh|2(x)χ(x) dx

+12

∑e∈EK

∫e

α [∂nkh]e(τ )χ(τ ) dτ).

(6.41)

34

Proposition 6.10. If Assumption 4.4 is satisfied, the following estimate holds for theindicators ηLK defined in (6.5), K ∈ Th,

ηLK ≤ c(f)(‖u∗ − uh‖W 1,ρ(ωK)d + ‖k∗ − kh‖W 1,ρ(ωK)

)+ c

∑κ⊂ωK

εLκ , (6.42)

for a constant c(f) only depending on the data f .

These estimates are no longer optimal according to the standard criteria. But thisseems unavoidable for non polynomial coefficients, even in the linear case of coefficientsonly depending on the space variable x, see [41, §3.2]. Moreover, if Eh denote the full error

Eh = ‖u− u∗‖W 1,ρ(Ω)d + ‖p− p∗‖Lρ(Ω) + ‖k − k∗‖W 1,ρ(Ω)

+ ‖u∗ − uh‖W 1,ρ(Ω)d + ‖p∗ − ph‖Lρ(Ω) + ‖k∗ − kh‖W 1,ρ(Ω),(6.43)

we observe from Proposition 6.1 and Corollary 6.2, Proposition 6.3 and Corollary 6.5 thatit admits the upper bound

Eh ≤ c ηm + c′( ∑K∈Th

(ηρK + hρK ‖f − fh‖

ρLρ(K)d

)) 1ρ

, (6.44)

which is fully optimal. On the other hand, it follows from Propositions 6.6, 6.8 and 6.10that it admits the lower bound

Eh ≥ c ηm + c′( ∑K∈Th

ηρK) 1ρ − c′′

( ∑K∈Th

(hρK ‖f − fh‖

ρLρ(K)d

+ (εSK)ρ + (εLK)ρ)) 1

ρ

, (6.45)

and the lack of optimality here comes from the εSK and εLK . However, since it follows fromLemmas 6.7 and 6.9 that a local bound for these terms can be computed explicitly, theadaptivity process can be performed in the usual way with the further requirement to becautious for the K such that εSK and εLK are not negligible with respect to ηSK and ηLK .

Another difficulty comes from the fact that the W−1,ρ(Ω)-norm which appears in thedefinition of ηm is not local. However, when setting

ηmK = ‖µ(kh)∇uh‖Lρ(K)d×d + ‖µ(kh)12 ∇uh‖2Lρ(K)d×d , (6.46)

and denoting by T `h the set of elements of Th which are contained in Ω`, we observe from(6.8) that

ηm ≤( ∑K∈T `

h

(ηmK )ρ) 1ρ . (6.47)

So these ηmK can reasonably be used in the strategy proposed in Section 2.

Remark 6.11. When replacing ν∗(·, kh) by νh in the discrete problem (5.3), as suggestedin Remark 5.1, it seems more natural to define the indicators ηSK and ηLK by

ηSK = hK ‖fh + div(νh(·)∇uh

)− (uh · ∇)uh − grad ph‖Lρ(K)d

+∑e∈EK

h1ρe ‖[νh(·) ∂nuh]e‖Lρ(e)d + ‖divuh‖Lρ(K),

ηLK = hK ‖νh(·) |∇uh|2 + α∆kh‖Lρ(K) +∑e∈EK

h1ρe ‖[α∂nkh]e‖Lρ(e).

(6.48)

All the estimates of this section remains valid in this case with only a further term, equalto(∑

K∈Th

((εSK)ρ + (εLK)ρ

)) 1ρ , in the right-hand side of (6.22) and (6.28).

35

7. Numerical experiments.

According to the adaptivity strategy proposed in Section 2, we must now solve thediscrete problem (5.3) for the sequence of subdomains Ωnt and Ωn` , provided with thetriangulation T nh . Since problem (5.3) is nonlinear, we also use the algorithm studied in[11] in a slightly different case (see also [42]), i.e. for a given k0

h in Yh, we iteratively solvethe uncoupled problems

Find (umh , pmh ) in Xh ×Mh such that

∀vh ∈ Xh,

∫Ω

ν∗(x, km−1h )∇umh : ∇vh dx+

∫Ω

(umh · ∇)umh · vh(x) dx

−∫

Ω

(div vh)(x)pmh (x) dx = 〈f ,vh〉,

∀qh ∈Mh, −∫

Ω

(divumh )(x)qh(x) dx = 0,

(7.1)

Find kmh in Yh such that

∀χh ∈ Yh, α

∫Ω

grad kmh · gradχh dx =∫

Ω

ν∗(x, km−1h ) |∇umh |2 χh(x) dx. (7.2)

The convergence of this algorithm is proved in [11, Th. 2] only under the assumptionthat the data f are small enough, however it seems likely in our case. The nonlinearterm (umh · ∇)umh in problem (7.1) is handled via the standard characteristics method,introduced in [37].

The numerical experiments that we present are performed on the code FreeFem++,see [23]. They correspond to the standard backward step problem. The domain Ω isL-shaped, given by

Ω =]− 4, 10[×]0, 1[ \ ]− 4, 0]×]0,12

]. (7.3)

The data f are equal to zero but the boundary conditions are now that of problem (1.1),i.e.

u = g on ∂Ω,

k = 0 on ∂Ω,(7.4)

where the function g is zero on all edges of ∂Ω except on the vertical edges contained inthe lines x = −4 and x = 10, where it is given by

g(−4, y) =(16(y − 1

2)(1− y), 0

),

12≤ y ≤ 1,

g(10, y) =(2y(1− y), 0

), 0 ≤ y ≤ 1.

(7.5)

The parameters ν0, ν1, ν2, α and ε satisfy the following conditions

ν1 = 10 ν0, ν2 = 1, α = 10−3, ε = 10−20. (7.6)

36

The next figures deal with the part of the flow between the lines x = −1 and x = 5. Figure2 presents, for ν0 = 10−4 and from top to bottom,• the final decomposition of Ω into Ωt (in light blue) and Ω` (in dark blue),• the final triangulation,• and the curves of isovalues of the stream function associated with the velocity uh.Figure 3 present the same quantities for ν0 = 10−5.

From these two figures, it can be observed that, when ν0 decreases, the size of Ωtand the density of triangles in this Ωt increase, which seems in good coherence with thebehaviour of the flow.

IsoValue012

Omega-T

Figure 2. Decomposition, triangulation and streamlines of the flow for ν0 = 10−4

37

IsoValue012

Omega-T

Figure 3. Decomposition, triangulation and streamlines of the flow for ν0 = 10−5

38

8. Conclusions.

As already said in the introduction, high Reynolds number flows are not turbulentin the whole domain. Moreover, any turbulence model requires additional computationalcost, in particular Reynolds Averaged Navier–Stokes (RANS) models such as the k − εsystem and its by-products. It is also known that usually direct numerical simulationsfail in real situations and therefore using the model is essential even in dimension 2 or foraxisymmetric cases (see [30]). Thus, our algorithm is useful to select turbulence regionsand reduce the computational cost.

Though the turbulence model that we have considered in this work is more a mathe-matical game than a real turbulence one, we stress that it contains the main mathematicalfeatures of a realistic RANS model. Our mathematical framework confirms that it isharder to analyze than the standard Navier–Stokes equations. Regularity assumptions arerequired to obtain a priori and a posteriori error estimates. Moreover, uniqueness is onlyproved in the two-dimenional case, with a smallness assumption on the source term whichis standard for the Navier–Stokes equations. Of course, the corresponding evolution equa-tions could have no steady state although the aim of the turbulence model is to computestatistical mean values of the real field. Nevertheless it is observed in [30] that such amodel evoluates quickly to a steady state for a real physical situation; but this remains anopen problem in the general case.

The numerical simulations in this present work show a good and stable numericalbehavior of the selective algorithm introduced in the paper. Convergence is observed andin the case of the classical backward facing step, we get the usual structure. This makes ourregularity assumptions likely. The concentration of finite elements observed in some regionshowever could be due to the lack of dissipation of the turbulence and near the wall to aboundary layer and the lack of wall laws. Indeed, using only one closure equation as we didis not sufficient: The mixing length and/or the turbulent dissipation must be parametrizedto get a realistic scale separation. Nevertheless, the analysis and the numerical simulationsthat we performed make our selective algorithm promising. It should now be tested over arealistic case, though this was not the aim of this first introducing and theoretical paper.

Acknowledgement: We are very grateful toward our colleague B. Mohammadi for hishelp concerning the derivation of the model. This work was partially supported by theMarie Curie EIF Programme of the European Union and also by the Programa Nacionalde I+D+I del Estado Espanol, Proyecto MTM2006-127575.

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